"Find f'(a) for f(x)=(1+2x)/(1+x)"
I got 2, but the answer is 1/(1+a)^2. How do I get that??
Differentiate f(x) to get f'(x), then
substitute a for x.
Let u = 1 + x
Then you want the derivative of (
(2u-1)/u. That is the derivative of -1/u (since the derivative of 2u/u is 0). The derivative is therefore is u^-2 = 1/(1+x)^2
That becomes 1/(1+a^2)when a is substituted for x.
Did you get f'(x) = 2 by simply taking the derivative of the top over the derivative of the bottom ??
my oh my !!
Thank you drwls!
Reiny - yes I did! I only realized I did that after I posted my question. Wasn't thinking for a moment. ;)
To find the derivative of a function, you can use the quotient rule. The quotient rule states that if you have a function of the form f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative f'(x) can be calculated as follows:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Now, let's apply the quotient rule to find the derivative of f(x) = (1 + 2x) / (1 + x):
f'(x) = [(1 + 2x)' * (1 + x) - (1 + 2x) * (1 + x)'] / [(1 + x)^2]
To simplify this expression, let's find the derivatives of the individual terms:
(1 + 2x)' = 2 (since the derivative of a constant is zero, and the derivative of 2x is 2)
(1 + x)' = 1 (since the derivative of a constant is zero, and the derivative of x is 1)
Substituting these values into the expression, we get:
f'(x) = (2 * (1 + x) - (1 + 2x) * 1) / (1 + x)^2
= (2 + 2x - 1 - 2x) / (1 + x)^2
= 1 / (1 + x)^2
So, you are correct! The derivative of f(x) = (1 + 2x) / (1 + x) is indeed 1 / (1 + x)^2.